What it takes to cover a region with no gaps and no overlaps

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What to assume they know

Tangrams: rearranging pieces without changing the area — the tangram, and covering with nothing bare and nothing doubled
Rows, columns and the size of a rectangular grid
Multiplication, and telling odd numbers from even
Knowing that if a total splits into pairs, that total must be even

What they should be able to do

State what it means for a region to be tiled by a given tile
Decide whether a rectangular grid can be tiled by 2 × 1 tiles, and give a general strategy when it can
Use a counting argument to rule out a tiling
Explain why counting alone cannot settle the fourteen-square regions
Construct the two-colour version of a tiling problem
Explain why a 2 × 1 tile must always cover one square of each colour
Use a colour-count mismatch to prove that no tiling exists
Name shapes whose copies tile the whole plane, and say why hexagons appear in nature

Where it usually goes wrong

"I tried for ten minutes and it did not work, so it cannot be done." That is the belief the whole section is aimed at. Failing to find an arrangement is not the same as showing none exists, and the colouring is what closes the gap.
"An even number of squares means it can be tiled." All three fourteen-square regions have an even count, and one of them is impossible. Evenness is necessary, not sufficient — say both halves of that.
"The colours are really there on the grid." They are not. The colouring is something the solver invents and lays over the problem, and it works because it is chosen so that no tile can avoid it.
"Colouring only rules tilings out." It rules them out; it never rules one in. Matching colour counts leaves the question open, which is exactly the position the two p.158 regions are in.
"A tile has to stay the way it is drawn." Rotation is allowed and the chapter says so; the two orientations are drawn and named on p.157.
"Every regular polygon tiles the plane." The chapter names three that do — squares, equilateral triangles and regular hexagons — and shows only those three. It does not claim the list is complete, and neither should the explanation.
"Tiling is finished mathematics." The chapter says the opposite: one of the four pictured tilings dates from 2023, and it calls the field active. Keep that ending.

Questions to check understanding

Decide whether a stated m × n grid can be tiled by 2 × 1 tiles and justify the answer
Complete the counting justification for the 5 × 7 grid (the task on p.158)
Give a general tiling strategy for an even-by-even grid, and for an even-by-odd one (the tasks on p.158)
Given a grid with one square removed, decide tileability and defend the answer (the tasks on p.158 and p.159)
Colour a given region and use the counts to prove no tiling exists (the argument on p.160)
Find another single square whose removal makes a 5 × 3 grid impossible to tile (the Math Talk task on p.160)
Decide whether a stated region can be tiled by a stated tile (Figure it Out questions 1 and 2, Part II, p.160)
Name regular polygons whose copies tile the plane
The tiling bullet of the SUMMARY (Part II, p.163) is the statement the chapter expects back

Examples worth working on the board

The 4 × 6 grid (Part II, §6.2, p.157). Checked against the printed page: a hatched rectangle of unit squares, four rows deep and six columns across, with the chapter naming rows before columns. Beneath it the two tile orientations are drawn and captioned, one standing and one lying. A completed tiling is printed further down the page with the joins picked out in green.
The 5 × 7 count (Part II, §6.2, pp.157–158). Inputs: 5 × 7 = 35 cells in all, and every tile takes exactly two of them. The chapter supplies both, then prints a line asking the reader to finish the justification. 35 is odd, and no number of twos makes an odd total.
The even-by-even strategy (Part II, §6.2, p.158). Checked against the printed page: a schematic grid with dashed continuation marks and an arrow down the left side labelled as an even number of tiles, showing each column filled by standing tiles.
The two fourteen-square regions (Part II, §6.2, p.158). Checked against the printed page. The first is a 5 × 3 grid with the top-right unit square taken out. The second is a 5 × 3 grid with the middle square of the second row taken out. Both have fourteen squares. The chapter asks about each under a Math Talk flag and prints no answer.
Fig. 6.13 (Part II, §6.2, p.159). Checked against the printed page: a 5 × 3 grid with the middle square of the top row taken out — fourteen squares again. This is the one that resists.
Fig. 6.14 (Part II, §6.2, p.159). Checked against the printed page: a two-by-two layout. Top left, the plain hatched region of Fig. 6.13; top right, the same region with its squares coloured alternately black and white; bottom left, the two plain tile orientations; bottom right, the same two tiles each carrying one black cell and one white cell.
The colour counts (Part II, §6.2, p.160). Inputs printed by the chapter: the coloured region holds 8 white squares and 6 black. Worked through on the printed page of p.159: in a 5 × 3 grid coloured so that neighbours differ and the corners are white, there are 8 white squares and 7 black; Fig. 6.13 removes a black one, leaving 8 and 6. Every tile takes one of each, so a covered region would need the two counts to match. They do not, so no arrangement can exist.
The two extra regions (Part II, §6.2, "Figure it Out", p.160). Checked against the printed page. Question 1 gives a twelve-square region — two columns wide for its top two rows, then four columns wide for its bottom two — and an L-shaped tile of three squares. Question 2 gives an 8 × 8 grid with two opposite corner squares taken out, sixty-two squares in all, and the plain 2 × 1 tile. See Notes: both answers are computed here, neither is printed.
Tilings of the plane (Part II, §6.2, pp.160–162). Checked against the printed page: a blue 3 × 4 block of squares with a cartoon figure carrying one more; a yellow patch of equilateral triangles; a blue patch of regular hexagons; then four pictorial tilings lettered (a) to (d) — an outlined pattern of squares and octagon-like cells, a blue-and-grey interlocking pattern, a fish tiling in red and green, and an interlocking tiling of horse-and-rider figures in yellow, green and dark blue on a black ground. The chapter credits Escher, gives his dates as 1898–1972, and dates the discovery of tiling (b) to 2023, which it calls recent. It closes with a bee and a honeycomb photograph.

Figures to have open

A 4 × 6 grid with draggable 2 × 1 tiles, able to be shown moving so more than one tiling can be shown. Standard schematic.
The three fourteen-square regions (the two on p.158 and Fig. 6.13), drawn to match the printed removals exactly — which square is missing is the entire point.
The same region in plain and two-colour versions, side by side, with a running tally of black and white. This is the topic's key image.
A 2 × 1 tile that keeps one black and one white cell however it is turned.
The 8 × 8 grid with two opposite corners removed, colourable.
Plane tilings by squares, equilateral triangles and regular hexagons, each extending past the frame edge. Redraw; the book's are decorative colour graphics.
A honeycomb photograph. The book prints one on p.162; source a fresh image.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 6 "Constructions and Tilings", §6.2 "Tiling", p.157 — the definition, the 4 × 6 grid, the two tile orientations, and the 5 × 7 count
Same part, same chapter, §6.2, p.158 — the justification prompt, the even-even and even-odd and odd-odd questions, and the two fourteen-square regions
Same part, same chapter, §6.2, p.159 — Fig. 6.13, the colouring idea, and Fig. 6.14
Same part, same chapter, §6.2, p.160 — the colour-count argument, the follow-up Math Talk task, and the "Figure it Out" questions 1 and 2
Same part, same chapter, §6.2, "Tiling the Entire Plane", pp.160–162, including the Escher tilings (a)–(d), the 2023 remark and the honeycomb
Same part, same chapter, SUMMARY, p.163, the tiling bullet
Backward pointers: Tangrams: rearranging pieces without changing the area, and Regular hexagons, and why the angles round a point must total 360° for the angles-round-a-point criterion that decides which regular polygons can meet edge to edge